EXAMPLES.-1. The altitude of a cylinder is 12 feet, and the diameter of its base 3 feet; required the solidity?

First, 3 x3.14159279.4247781 circumference of the base. Aft. 287.

Then,

3 x 9.4247781

4

=7.0685836=area of the base. Art.

=

288.: 7.0685836 × 12=84.8230032 cubic feet the solidity required.

2. The altitude is 20 feet, and the circumference of the base 20 feet; required the solid content of the cylinder? Ans. 636.64 feet.

3. The diameter of the base is 4 feet, and the altitude 9 feet; required the solidity of the cylinder?

293. To find the solid content of a cone.

RULE. Find the solidity of a cylinder of the same base and altitude with the given cone, by the last rule; one third of this will be the solid content of the cone o.

EXAMPLES 1. The circumference of the base of a cone is 12 feet, and its altitude 10 feet; required the solid content ?

Whence 11.459154 × 10=114.59154=solidity of the circumscribing cylinder. Art. 292.

114.59154

Lastly,

3

33.19718 cubic feet the solidity of the

cone.

For the foundation of the rule sée Euclid 10. 12. Let a=the axis of a cone, d=the semidiameter of its base, then (Euclid 47. 1.) a2+d2=the slant height of the cone; and if the slant height be multiplied into the circumference of the base, the product will be the convex superficies of the cone, to which, adding the area of the base, the sum will be the whole external superficies. Rules for finding the superficies and solidities of the several sections of a prism, pyramid, cone, cylinder, sphere, &c. may be found in Mr. Bonnycastle's excellent Introduction to Mensuration, a work which cannot be too highly commended.

2. The altitude is 12, and the diameter of the base 3; required the solidity of the cone? Ans. 28.2743344.

3. The area of the base is 20, and the altitude 14; required the solid content of the cone?

294. To find the solid content of a sphere.

RULE. Find the solidity of a cylinder, of which the altitude, and the diameter of its base, are each equal to the diameter of the given sphere; two thirds of this will be the solidity of the sphere.

Euclid has proved that "spheres are to each other in the triplicate ratio of their diameters" (18. 12.); but this is the only property of the sphere to be found in the Elements. We are beholden to Archimedes for the most part of our original information on this subject; the above rule, which was taken from his treatise "on the sphere and cylinder," may be easily demonstrated by indivisibles," "the method of increments," "fluxions," and some other modern methods of computation; but I believe it cannot be effected by elementary Geometry.

The superficies of a sphere is equal to the convex surface of its circumscribing ●ylinder; it is likewise equal to four times the area of a great circle of the sphere.

If the diameter of a sphere be 2, then will the circumference of a great

superficies... 9.57454
solidity

.....

2.53615

Hence the superficial and solid content of a solid, similar to any of the above, may be readily obtained, its side being given; the superficies being as the squares (Euclid 20. 6.), and the solidities as the cubes (cor, &. 12.) of the homologous sides.

A a $

EXAMPLES.-1. The diameter of a sphere is 3 feet; required

its solidity?

First, 3 x3.1415927=9.4247781-circumf. of the cylin

Thirdly, 7.0685836x3=21.2057508=the solidity of the cylinder. Art. 292.

=

Lastly, of 21.2057508-14.1371672 cubic feet the soli dity of the sphere.

2. The diameter of a sphere is 17 inches; required its solidity? Ans. 1.48868 cubic feet.

3. If the earth be a perfect sphere of 8000 miles diameter, how many cubic miles of matter does it contain?

PART IX.

TRIGONOMETRY,

HISTORICAL INTRODUCTION.

a

TRIGONOMETRY is a science which teaches how to determine the sides and angles of triangles, by means of the relations and properties of certain right lines drawn in and about the circle; it is divided into two kinds, plane and spherical, the former of which applies to the computation of plane rectilineal triangles, and the latter to triangles formed by the intersections of great circles, on the surface of a sphere.

This science is justly considered as an important link connecting theoretical Geometry with practical utility, and making the former conducive, and subservient to the latter. Geography, Astronomy, Dialling, Navigation, Surveying, Mensuration, Fortification, &c. are indebted to it, if not for their existence, at least for their distinguishing perfections; and there is scarcely any branch of Natural Philosophy, which can be successfully cultivated without, the assistance of Trigonometry.

We are in possession of no documents that will warrant us even to guess at the period when Trigonometry took its rise; but there can be do doubt that it must have been invented not very long after the flood. The earliest inhabitants of Chaldæa and Egypt were acquainted with Astronomy, which

a The name is derived from rgus three, yovos a corner, and μirgiw to measure. The objects of Trigonometry are the sides and angles only, whatever respects the areas of triangles belongs to Geometry.

(admitting it to have been at that time merely an art, and in its rudest state) would still require the aid of some method similar to Trigonometry to make it of any benefit to mankind.

We may reasonably suppose that the ancient Greeks cultivated Trigonometry, in common with Geometry and Astronomy; but none of their writings on the subject have been preserved. Theon, in his Commentary on Ptolemy's Almagest, mentions a work consisting of twelve books on the chords of circular arcs, written by Hipparchus, an Astronomer of Rhodes, A. C. 130. This work is believed by the learned to have been a treatise on the ancient Trigo

Theon, a respectable mathematician and philosopher, and president of the Alexandrian school, flourished A. D. 370. He was not more famous for his acquirements in science, than for his veneration of the DEITY, and his firm belief in the constant superintendence of divine providence; he rècommends meditation on the presence of God, as the most delightful and useful 'employment, and proposed, that in order to deter the profligate from committing crimes, there should be written at the corner of every street, Remember God SEES THEE, O SINNER. Dr. Simson, in his notes on the Elements of Euclid, has ascribed most of the faults in that book to Theon, without mentioning on what authority he has done so.

c Hipparchus was born at Nice, in Bithynia: here, and afterwards at Rhodes and Alexandria, his astronomical observations were made. He discovered that the interval between the vernal and autumnal equinox is longer by 7 days than that between the autumnal and vernal; he was the first who arranged the stars into 49 constellations, and determined their longitudes and apparent magnitudes; and his labours in this respect were considered so valuable, that Ptolemy has inserted his catalogue of the fixed stars in his Almagest, where it is still preserved. He also discovered the precession of the equinoxes, and the parallax of the planets; and, after the example of Thales, and Sulpicius Gallus, foretold the exact time of eclipses, of which he made a calculation for 600 years. He determined the latitude and longitude, and fixed the first meridian at the 'Fortunate Insulæ, or Canary Islands; in which particular he has been followed by most succeeding geographers. Astronomy is particularly indebted to him for collecting the detached and scattered principles and observations of his predecessors, arranging them in a system; thereby laying that rational and solid foundation, upon which succeeding astronomers have built a most sublime and magnificent superstructure. Of the several works said to have been written by him, his Commentary on the Phænomena of Aratus is the only one that remains.